152 R. FUJI s A WA. 



Jncobi adds " hi loi (jcucral de la compomtion iJc cfs expressions est 

 aisée à saisir,'' and iVirther remarks that we sliaJl have aiialoofoiis 

 lorinula' l^y usiiio', instead of the ditterentia] coefficients of A and B, 

 those of 



and 



V.?'-'(l - -r-^Xl - F-.r') ' /(±^ ./•-)( I - /.-'■') • 



x^ 



The o^eneral form of these expressions is, however, by no means easy 

 to infer from the ])articular cases just f^iven, and I have tried to trace 

 among- the writings of Jacohi, the steps wliich might have led him to 

 these expressions, but without success. 



The present memoir is divided into two ])arts. In the first part, 

 the miiltip]ication-formn]a' of elliptic functions are derived from Abel's 

 theorem for the elliptic integral of the first kind. It will be seen that 

 one of the results arrived at is the general formula in question. It 

 appears, however, liighly improbable th.at flacobi obtained liis formulae 

 in this way. 



In the ])aper just alluded to, pJacobi gives, also without demon- 

 stration, the partial differential equation satisfied by the numerators and 

 denominator of the nudtiplication-formula?. This partial differential 

 e(|uation has since been obtained by lîetti,* Cayley,** Briot et 

 r)()n(|uet,'j' and others ; but the final results to ])e obtained by 

 applying it (o the actual evahiation of the numerical constants involved 

 in the multiplication-formuke, has not, to my knowledge, hitherto 

 been developed with much completeness or success. 



In the second part, Jacobi's partial differential equation is derived 

 in a manner which is most probably the one followed by Jacobi 



* Betti, Annali di Mateniatica, Vol. IV, p. 32. 



** Cayley, Cambridge and Dublin Mathematical Journal, Vol. II, pp. 250-26G. 



t Briot et Bouquet, Theorie des Fonctions Elliptiques, p. 529. 



